On certain classes of first Baire functionals

Abstract We investigate first Baire functionals on the dual ball of a separable Banach space $$X$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>X</mml:mi> </mml:math> which are pointwise limit of a sequence of $$X$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>X</mml:mi> </mml:math> whose closed span does not contain any copy of $$\ell_1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>ℓ</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:math> (or has separable dual). We propose an example of a $$C(K)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>C</mml:mi> <mml:mo>(</mml:mo> <mml:mi>K</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> space where not all such first Baire functionals exhibit this behavior. As an application, we study a quantitative version, in terms of descriptive set theory, of family a separable Banach spaces with this peculiarity.